College Algebra, Chapter 4, 4.1, Section 4.1, Problem 26

A quadratic function $f(x) = 5x^2 + 30x + 4$.

a.) Find the quadratic function in standard form.


$
\begin{equation}
\begin{aligned}

f(x) =& 5x^2 + 30x + 4
&&
\\
\\
f(x) =& 5 (x^2 + 6x ) + 4
&& \text{Factor out 5 from the $x$-term}
\\
\\
f(x) =& 5 (x^2 + 6x + 9) + 4 - (5)(9)
&& \text{Complete the square: add $9$ inside the parentheses, subtract $(5)(9)$ outside}
\\
\\
f(x) =& 5 (x + 3) - 41
&& \text{Factor and simplify}

\end{aligned}
\end{equation}
$


The standard form is $f(x) = 5 (x + 3) - 41$.

b.) Draw its graph.







c.) Find its maximum or minimum value.

Based from the graph in part (b), since the graph opens upward the minimum value of $f$ is $f(-3) = -41$.